Line game-perfect graphs

The $[X,Y]$-edge colouring game is played with a set of k colours on a graph G with initially uncoloured edges by two players, Alice (A) and Bob (B). The players move alternately. Player X∈,B\ has the first move. Y∈,B,-\. If Y∈,B\, then only player Y may skip any move, otherwise skipping is not allowed for any player. A move consists of colouring an uncoloured edge with one of the k colours such that adjacent edges have distinct colours. When no more moves are possible, the game ends. If every edge is coloured in the end, Alice wins; otherwise, Bob wins. The $[X,Y]$-game chromatic index χ[X,Y]'(G) is the smallest nonnegative integer k such that Alice has a winning strategy for the $[X,Y]$-edge colouring game played on G with k colours. The graph G is called line $[X,Y]$-perfect if, for any edge-induced subgraph H of G, \[χ[X,Y]'(H)=ω(L(H)),\] where ω(L(H)) denotes the clique number of the line graph of H. For each of the six possibilities (X,Y)∈,B\×,B,-\, we characterise line $[X,Y]$-perfect graphs by forbidden (edge-induced) subgraphs and by explicit structural descriptions, respectively.